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. Let H={z=x+iy;y>0} be a hyperbolic plane with the Poincare metric ds2=(dx2+dye)/y2 of constant curvature -1. The group PSL(2, R) acts on H as the group of orientation-preserving isometrics. A hyperbolic element ƒÁ of PSL(2, R) has two fixed points in R•3⁄4{•‡}; the repelling fixed point ƒ¿ and the attracting one ƒÀ. The geodesic aƒÁ connecting ƒ¿ and ƒÀ is called the axis of ƒÁ. Let ƒ¡ be a Fuchsian group in PSL(2, R) and ƒÓ=ƒÓƒ¡: H• ̈H/ƒ¡ be the natural projection on the quotient space. Then the equivalence classes of axes {ƒÅ(aƒÁ); ƒÅƒ°ƒ¡} of hyperbolic elements of ƒ¡ and the closed geodesics on H/ƒ¡ (which include a kind of geodesic segments for some cases, see • ̃1) are in one-to-one correspondence under the map induced by ƒÓ: aƒÁƒÓ(aƒÁ). The purpose of the present paper is to show that a closed geodesic with some self-intersections cannot be too short. To state our main theorem we first give the following condition imposed on hyperbolic elements ƒÁ of PSL(2, R): (•‡) There exists a Fuchsian group ƒ¡ containing ƒÁ and another element ƒÂ in such a way that ƒÂ does not preserve the axis aƒÁ of ƒÁ (that is, aƒÁ•‚ƒÂ(aƒÁ)) and that aƒÁ and ƒÂ(aƒÁ)=aƒÂƒÁƒÂ-1 intersect each other.

The lengths of the closed geodesics on a Riemann surface with self-intersections